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Towards Pósa's Conjecture for $3$-graphs

Debmalya Bandyopadhyay, Allan Lo, Richard Mycroft

Abstract

We prove that every $3$-graph $H$ on $n$ vertices with minimum codegree $δ_2(H) \geq 7n/9 + o(n)$ contains the square of a tight Hamilton cycle. This strengthens a theorem of Bedenknecht and Reiher that $δ_2(H) \geq 4n/5 + o(n)$ is sufficient. The central novelty of our arguments is an improved understanding of the connectivity structure of $3$-graphs with large minimum codegree.

Towards Pósa's Conjecture for $3$-graphs

Abstract

We prove that every -graph on vertices with minimum codegree contains the square of a tight Hamilton cycle. This strengthens a theorem of Bedenknecht and Reiher that is sufficient. The central novelty of our arguments is an improved understanding of the connectivity structure of -graphs with large minimum codegree.

Paper Structure

This paper contains 14 sections, 37 theorems, 55 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Key ideas and definitions
  3. Preliminary results and notation
  4. Extremal construction
  5. Turán density
  6. Probabilistic tools
  7. Tight Connectivity
  8. Every vertex is in at most two tight components
  9. There are at most two tight components
  10. There is only one tight component
  11. Hypergraph Regularity
  12. Connecting Lemma
  13. Absorption
  14. Path Cover Lemma

Key Result

Theorem 1.1

For all $\alpha > 0$ there exists $n_0 \in \mathbb{N}$ such that every $3$-graph $H$ on $n \geq n_0$ vertices with $\delta_2(H) \geq (7/9 + \alpha) n$ contains the square of a tight Hamilton cycle.

Figures (3)

  • Figure 1: Edges and components in Lemma \ref{['Lma: two tight components']}
  • Figure 2: A tight path $v_1v_2\dots v_6$ with $\phi(v_1v_2v_3) = \phi(v_3v_4v_5) = r$ and $\phi(v_2v_3v_4) = \phi(v_4v_5v_6) = b$ mentioned in Lemma \ref{['lma: no alt tight path length 4']}
  • Figure 3: A tight path $v_1v_2\dots v_7$ with $\phi(v_1v_2v_3) = \phi(v_3v_4v_5) = \phi(v_4v_5v_6)= r$ and $\phi(v_2v_3v_4) = \phi(v_5v_6v_7) = b$ mentioned in Lemma \ref{['lma: no weird tight path length 5']}

Theorems & Definitions (73)

  • Theorem 1.1
  • Conjecture 1.2
  • Proposition 1.3
  • Lemma 1.4
  • Lemma 1.5: Connecting Lemma
  • Lemma 1.6: Absorption Lemma
  • Lemma 1.7: Path Cover Lemma
  • proof : Proof of Theorem \ref{['thm: ourthm']}
  • proof : Proof of Proposition \ref{['construct']}
  • Theorem 2.1: cf. Keevash, Mubayi-Zhao
  • ...and 63 more